PSI - Issue 2_A

Cherny S.G. et al. / Procedia Structural Integrity 2 (2016) 2479–2486 Cherny S.G., Lapin V.N. / Structural Integrity Procedia 00 (2016) 000–000

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Fig. 3. Poiseuille velocity profiles for Newtonian (left) and Herchel-Bulkley (right) fluids.

In case of one-dimensional power law ( τ 0 ≡ 0) fluid flow inside the fracture the apparent viscosity (9) was used by Ouyang et al. (1997) and Rungamornrat et al. (2005). The expressions (2) and (9) are combined with (1) to obtain the Reynolds equation (3) with apparent viscosity (9) instead of the constant viscosity µ . Like in case of Newonian fluid the FEM gives the system of equations similar to (7). But because the coe ffi cient a in (4) now depends on the pressure the matrix K ( P ) and the vector Q ( P ) also do and so the equations in the system (7) are nonlinear. The following auxiliary iteration process is introduced to solve the system and to calculate the pressure distribution 1. s = 0 : The pressure from the previous step of the fracture propagation is taken as the initial solution P s = P n ; 2. The coe ffi cients a s is calculated at each point using (4); 3. The interim pressure P is calculated from the solution of (7) with K ( P s ), Q ( P s ); 4. The pressure distribution at the next iteration is calculated using the relaxation procedure P s + 1 = P ( r ) + P s (1 − r ) , where r ( s ) = r max || P s || || P s − P || ; (10) 5. s = s + 1. The iterations 2–4 are repeated untill the condition || P s − P || || P s || < ε c is fullfield. The values r max = 0 . 1 , and ε c = 10 − 4 are used in the current calculations. To take the fluid compressibility into account one should add the fluid density ρ and rewrite the continuity equation (1) in the form ∂ W ρ ∂ t + ∇ · ( ρ q ) = 0 . (11) In case of low compressible fluid the density obeys the law ρ ( P ) = ρ 0 (1 + C 0 P ) , (12) where C 0 is the compressibility coe ffi cient. It allows to exclude the density from the (11) and to obtain the modified mass conservation equation ∂ ∂ t [ W (1 + C 0 P )] + ∇ · [(1 + C 0 P ) q ] = 0 . (13) As in case of Newtonian incompressible fluid (Sec. 2.2) the equations (11) and (2) are combined into the equation (3). But coe ffi cients of (3) depend on the pressure 2.4. Model of compressible fluid

(1 + C 0 P ) W 3 12 µ

∂ W (1 + C 0 P ) ∂ t .

, f =

(14)

a =

As in case of Hershel-Bulkley fluid (Sec. 2.3) the equations in the system (7) are nonlinear. The same iteration procedure is applied to obtain it’s solution. The only di ff erence is that at the steps 2 and 3 the coe ffi cients f ( P s ) and F ( P s ) should be also recalculated with regarding the pressure P s .